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Mirrors > Home > ILE Home > Th. List > apsym | Unicode version |
Description: Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
Ref | Expression |
---|---|
apsym | # # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 7874 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | cnre 7874 | . . . . . 6 | |
4 | 3 | ad3antrrr 484 | . . . . 5 |
5 | simplrl 525 | . . . . . . . . . . . 12 | |
6 | simplrl 525 | . . . . . . . . . . . . 13 | |
7 | 6 | ad2antrr 480 | . . . . . . . . . . . 12 |
8 | reaplt 8463 | . . . . . . . . . . . 12 # | |
9 | 5, 7, 8 | syl2anc 409 | . . . . . . . . . . 11 # |
10 | reaplt 8463 | . . . . . . . . . . . . 13 # | |
11 | 7, 5, 10 | syl2anc 409 | . . . . . . . . . . . 12 # |
12 | orcom 718 | . . . . . . . . . . . 12 | |
13 | 11, 12 | bitr4di 197 | . . . . . . . . . . 11 # |
14 | 9, 13 | bitr4d 190 | . . . . . . . . . 10 # # |
15 | simplrr 526 | . . . . . . . . . . . 12 | |
16 | simplrr 526 | . . . . . . . . . . . . 13 | |
17 | 16 | ad2antrr 480 | . . . . . . . . . . . 12 |
18 | reaplt 8463 | . . . . . . . . . . . 12 # | |
19 | 15, 17, 18 | syl2anc 409 | . . . . . . . . . . 11 # |
20 | reaplt 8463 | . . . . . . . . . . . . 13 # | |
21 | 17, 15, 20 | syl2anc 409 | . . . . . . . . . . . 12 # |
22 | orcom 718 | . . . . . . . . . . . 12 | |
23 | 21, 22 | bitr4di 197 | . . . . . . . . . . 11 # |
24 | 19, 23 | bitr4d 190 | . . . . . . . . . 10 # # |
25 | 14, 24 | orbi12d 783 | . . . . . . . . 9 # # # # |
26 | apreim 8478 | . . . . . . . . . 10 # # # | |
27 | 5, 15, 7, 17, 26 | syl22anc 1221 | . . . . . . . . 9 # # # |
28 | apreim 8478 | . . . . . . . . . 10 # # # | |
29 | 7, 17, 5, 15, 28 | syl22anc 1221 | . . . . . . . . 9 # # # |
30 | 25, 27, 29 | 3bitr4d 219 | . . . . . . . 8 # # |
31 | simpr 109 | . . . . . . . . 9 | |
32 | simpllr 524 | . . . . . . . . 9 | |
33 | 31, 32 | breq12d 3978 | . . . . . . . 8 # # |
34 | 32, 31 | breq12d 3978 | . . . . . . . 8 # # |
35 | 30, 33, 34 | 3bitr4d 219 | . . . . . . 7 # # |
36 | 35 | ex 114 | . . . . . 6 # # |
37 | 36 | rexlimdvva 2582 | . . . . 5 # # |
38 | 4, 37 | mpd 13 | . . . 4 # # |
39 | 38 | ex 114 | . . 3 # # |
40 | 39 | rexlimdvva 2582 | . 2 # # |
41 | 2, 40 | mpd 13 | 1 # # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1335 wcel 2128 wrex 2436 class class class wbr 3965 (class class class)co 5824 cc 7730 cr 7731 ci 7734 caddc 7735 cmul 7737 clt 7912 # cap 8456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-ltxr 7917 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 |
This theorem is referenced by: addext 8485 mulext 8489 ltapii 8510 ltapd 8513 apdivmuld 8686 div2subap 8709 recgt0 8721 prodgt0 8723 pwm1geoserap1 11405 absgtap 11407 geolim 11408 geolim2 11409 geo2sum2 11412 geoisum1c 11417 tanaddap 11636 egt2lt3 11676 sqrt2irraplemnn 12053 triap 13600 apdiff 13619 |
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